\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  -13.0 & +  9.00 x_{1} & -5.00 x_{2} & -9.00 x_{3} & -10.00 x_{4} & -3.00 x_{5} & +  5.00 x_{6} & +  6.00 x_{7} & +  4.00 x_{8}\\
 x_{10}   &  44.0  &   & -9.00 x_{2} & -7.00 x_{3} & -6.00 x_{4} & -5.00 x_{5} & -10.00 x_{6} & +  1.00 x_{7} & +  6.00 x_{8}\\
 x_{11}   &  -7.0  &   & -2.00 x_{2} & +  5.00 x_{3} & +  4.00 x_{4} & +  8.00 x_{5} & -6.00 x_{6} & +  8.00 x_{7} & + 10.00 x_{8}\\
 x_{12}   &  31.0 & -9.00 x_{1} & -10.00 x_{2} & -2.00 x_{3} & +  3.00 x_{4} & +  3.00 x_{5} & +  7.00 x_{6} & -10.00 x_{7} & +  6.00 x_{8}\\
 x_{13}   &  0.0 & -10.00 x_{1} & +  3.00 x_{2} & +  7.00 x_{3} & -8.00 x_{4} & +  5.00 x_{5} & +  6.00 x_{6} & +  5.00 x_{7} & +  9.00 x_{8}\\
 x_{14}   &  -5.0 & +  7.00 x_{1} & -10.00 x_{2} & +  5.00 x_{3} & -9.00 x_{4} & + 10.00 x_{5} & +  7.00 x_{6} & -6.00 x_{7} & +  4.00 x_{8}\\
\hline
z    &  0.0 & +  2.00 x_{1} & +  5.00 x_{2} & +  4.00 x_{3} & -2.00 x_{4} & -3.00 x_{5} & +  4.00 x_{6} & -5.00 x_{7} & -1.00 x_{8}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,14$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -9.00 y_{9} &    &   & +  9.00 y_{12} & + 10.00 y_{13} & -7.00 y_{14}\\
 y_{2}   &  1.0 & +  5.00 y_{9} & +  9.00 y_{10} & +  2.00 y_{11} & + 10.00 y_{12} & -3.00 y_{13} & + 10.00 y_{14}\\
 y_{3}   &  1.0 & +  9.00 y_{9} & +  7.00 y_{10} & -5.00 y_{11} & +  2.00 y_{12} & -7.00 y_{13} & -5.00 y_{14}\\
 y_{4}   &  1.0 & + 10.00 y_{9} & +  6.00 y_{10} & -4.00 y_{11} & -3.00 y_{12} & +  8.00 y_{13} & +  9.00 y_{14}\\
 y_{5}   &  1.0 & +  3.00 y_{9} & +  5.00 y_{10} & -8.00 y_{11} & -3.00 y_{12} & -5.00 y_{13} & -10.00 y_{14}\\
 y_{6}   &  1.0 & -5.00 y_{9} & + 10.00 y_{10} & +  6.00 y_{11} & -7.00 y_{12} & -6.00 y_{13} & -7.00 y_{14}\\
 y_{7}   &  1.0 & -6.00 y_{9} & -1.00 y_{10} & -8.00 y_{11} & + 10.00 y_{12} & -5.00 y_{13} & +  6.00 y_{14}\\
 y_{8}   &  1.0 & -4.00 y_{9} & -6.00 y_{10} & -10.00 y_{11} & -6.00 y_{12} & -9.00 y_{13} & -4.00 y_{14}\\
\hline
z    &  -0 & + 13.00 y_{9} & -44.00 y_{10} & +  7.00 y_{11} & -31.00 y_{12} &   & +  5.00 y_{14}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 6 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{9}   &  0.142604943115 & -0.04 y_{1} & +  0.66 y_{10} & -0.09 y_{7} & +  1.07 y_{12} & +  0.04 y_{8} & -0.05 y_{6}\\
 y_{2}   &  1.7373479796 & -0.61 y_{1} & +  5.95 y_{10} & +  0.52 y_{7} & +  6.50 y_{12} & -0.62 y_{8} & -0.02 y_{6}\\
 y_{3}   &  1.93723028639 & -0.50 y_{1} & + 17.22 y_{10} & -1.16 y_{7} & + 21.93 y_{12} & +  1.16 y_{8} & -0.44 y_{6}\\
 y_{4}   &  2.82169478227 & -0.19 y_{1} & + 17.29 y_{10} & -0.26 y_{7} & -8.16 y_{12} & -0.14 y_{8} & -1.24 y_{6}\\
 y_{5}   &  1.07944291879 & +  0.01 y_{1} & + 16.80 y_{10} & -1.15 y_{7} & + 14.46 y_{12} & +  1.48 y_{8} & -0.41 y_{6}\\
 y_{14}   &  0.0127500980777 & -0.02 y_{1} & -0.30 y_{10} & +  0.09 y_{7} & -1.11 y_{12} & -0.07 y_{8} & -0.00 y_{6}\\
 y_{13}   &  0.0372695174578 & +  0.05 y_{1} & +  0.38 y_{10} & -0.02 y_{7} & -0.71 y_{12} & -0.02 y_{8} & -0.05 y_{6}\\
 y_{11}   &  0.00431541781091 & -0.02 y_{1} & -1.09 y_{10} & +  0.01 y_{7} & +  0.05 y_{12} & -0.07 y_{8} & +  0.07 y_{6}\\
\hline
z    &  1.94782267556 & -0.76 y_{1} & -44.54 y_{10} & -0.58 y_{7} & -22.20 y_{12} & -0.38 y_{8} & -0.23 y_{6}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.764809729306 & +  0.04 x_{9} & +  0.61 x_{2} & +  0.50 x_{3} & +  0.19 x_{4} & -0.01 x_{5} & +  0.02 x_{14} & -0.05 x_{13} & +  0.02 x_{11}\\
 x_{10}   &  44.5355041193 & -0.66 x_{9} & -5.95 x_{2} & -17.22 x_{3} & -17.29 x_{4} & -16.80 x_{5} & +  0.30 x_{14} & -0.38 x_{13} & +  1.09 x_{11}\\
 x_{7}   &  0.577481365241 & +  0.09 x_{9} & -0.52 x_{2} & +  1.16 x_{3} & +  0.26 x_{4} & +  1.15 x_{5} & -0.09 x_{14} & +  0.02 x_{13} & -0.01 x_{11}\\
 x_{12}   &  22.2047861907 & -1.07 x_{9} & -6.50 x_{2} & -21.93 x_{3} & +  8.16 x_{4} & -14.46 x_{5} & +  1.11 x_{14} & +  0.71 x_{13} & -0.05 x_{11}\\
 x_{8}   &  0.375833660259 & -0.04 x_{9} & +  0.62 x_{2} & -1.16 x_{3} & +  0.14 x_{4} & -1.48 x_{5} & +  0.07 x_{14} & +  0.02 x_{13} & +  0.07 x_{11}\\
 x_{6}   &  0.229697920753 & +  0.05 x_{9} & +  0.02 x_{2} & +  0.44 x_{3} & +  1.24 x_{4} & +  0.41 x_{5} & +  0.00 x_{14} & +  0.05 x_{13} & -0.07 x_{11}\\
\hline
z    &  -1.94782267556 & -0.14 x_{9} & -1.74 x_{2} & -1.94 x_{3} & -2.82 x_{4} & -1.08 x_{5} & -0.01 x_{14} & -0.04 x_{13} & -0.00 x_{11}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.764809729306 & +  0.04 x_{9} & +  0.61 x_{2} & +  0.50 x_{3} & +  0.19 x_{4} & -0.01 x_{5} & +  0.02 x_{14} & -0.05 x_{13} & +  0.02 x_{11}\\
 x_{10}   &  44.5355041193 & -0.66 x_{9} & -5.95 x_{2} & -17.22 x_{3} & -17.29 x_{4} & -16.80 x_{5} & +  0.30 x_{14} & -0.38 x_{13} & +  1.09 x_{11}\\
 x_{7}   &  0.577481365241 & +  0.09 x_{9} & -0.52 x_{2} & +  1.16 x_{3} & +  0.26 x_{4} & +  1.15 x_{5} & -0.09 x_{14} & +  0.02 x_{13} & -0.01 x_{11}\\
 x_{12}   &  22.2047861907 & -1.07 x_{9} & -6.50 x_{2} & -21.93 x_{3} & +  8.16 x_{4} & -14.46 x_{5} & +  1.11 x_{14} & +  0.71 x_{13} & -0.05 x_{11}\\
 x_{8}   &  0.375833660259 & -0.04 x_{9} & +  0.62 x_{2} & -1.16 x_{3} & +  0.14 x_{4} & -1.48 x_{5} & +  0.07 x_{14} & +  0.02 x_{13} & +  0.07 x_{11}\\
 x_{6}   &  0.229697920753 & +  0.05 x_{9} & +  0.02 x_{2} & +  0.44 x_{3} & +  1.24 x_{4} & +  0.41 x_{5} & +  0.00 x_{14} & +  0.05 x_{13} & -0.07 x_{11}\\
\hline
z    &  -0.814829344841 & -0.10 x_{9} & +  8.25 x_{2} & +  2.14 x_{3} & +  1.91 x_{4} & -5.66 x_{5} & +  0.42 x_{14} & +  0.01 x_{13} & -0.24 x_{11}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.764809729306 & +  0.04 x_{9} & +  0.61 x_{2} & +  0.50 x_{3} & +  0.19 x_{4} & -0.01 x_{5} & +  0.02 x_{14} & -0.05 x_{13} & +  0.02 x_{11}\\
 x_{10}   &  44.5355041193 & -0.66 x_{9} & -5.95 x_{2} & -17.22 x_{3} & -17.29 x_{4} & -16.80 x_{5} & +  0.30 x_{14} & -0.38 x_{13} & +  1.09 x_{11}\\
 x_{7}   &  0.577481365241 & +  0.09 x_{9} & -0.52 x_{2} & +  1.16 x_{3} & +  0.26 x_{4} & +  1.15 x_{5} & -0.09 x_{14} & +  0.02 x_{13} & -0.01 x_{11}\\
 x_{12}   &  22.2047861907 & -1.07 x_{9} & -6.50 x_{2} & -21.93 x_{3} & +  8.16 x_{4} & -14.46 x_{5} & +  1.11 x_{14} & +  0.71 x_{13} & -0.05 x_{11}\\
 x_{8}   &  0.375833660259 & -0.04 x_{9} & +  0.62 x_{2} & -1.16 x_{3} & +  0.14 x_{4} & -1.48 x_{5} & +  0.07 x_{14} & +  0.02 x_{13} & +  0.07 x_{11}\\
 x_{6}   &  0.229697920753 & +  0.05 x_{9} & +  0.02 x_{2} & +  0.44 x_{3} & +  1.24 x_{4} & +  0.41 x_{5} & +  0.00 x_{14} & +  0.05 x_{13} & -0.07 x_{11}\\
\hline
z    &  -0.814829344841 & -0.10 x_{9} & +  8.25 x_{2} & +  2.14 x_{3} & +  1.91 x_{4} & -5.66 x_{5} & +  0.42 x_{14} & +  0.01 x_{13} & -0.24 x_{11}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{7} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  1.45095989356 & +  0.14 x_{9} & -1.19 x_{7} & +  1.87 x_{3} & +  0.50 x_{4} & +  1.36 x_{5} & -0.08 x_{14} & -0.03 x_{13} & -0.00 x_{11}\\
 x_{10}   &  37.8773997339 & -1.64 x_{9} & + 11.53 x_{7} & -30.57 x_{3} & -20.25 x_{4} & -30.09 x_{5} & +  1.30 x_{14} & -0.57 x_{13} & +  1.25 x_{11}\\
 x_{2}   &  1.11917886333 & +  0.17 x_{9} & -1.94 x_{7} & +  2.24 x_{3} & +  0.50 x_{4} & +  2.23 x_{5} & -0.17 x_{14} & +  0.03 x_{13} & -0.03 x_{11}\\
 x_{12}   &  14.9310017107 & -2.15 x_{9} & + 12.60 x_{7} & -36.51 x_{3} & +  4.92 x_{4} & -28.98 x_{5} & +  2.20 x_{14} & +  0.51 x_{13} & +  0.12 x_{11}\\
 x_{8}   &  1.0731800038 & +  0.07 x_{9} & -1.21 x_{7} & +  0.24 x_{3} & +  0.45 x_{4} & -0.08 x_{5} & -0.03 x_{14} & +  0.04 x_{13} & +  0.05 x_{11}\\
 x_{6}   &  0.248907051891 & +  0.06 x_{9} & -0.03 x_{7} & +  0.48 x_{3} & +  1.24 x_{4} & +  0.45 x_{5} & +  0.00 x_{14} & +  0.05 x_{13} & -0.07 x_{11}\\
\hline
z    &  8.42026230755 & +  1.27 x_{9} & -15.99 x_{7} & + 20.64 x_{3} & +  6.02 x_{4} & + 12.77 x_{5} & -0.96 x_{14} & +  0.27 x_{13} & -0.46 x_{11}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  2.21732101857 & +  0.03 x_{9} & -0.54 x_{7} & -0.05 x_{12} & +  0.75 x_{4} & -0.12 x_{5} & +  0.03 x_{14} & -0.00 x_{13} & +  0.01 x_{11}\\
 x_{10}   &  25.3771274463 & +  0.15 x_{9} & +  0.98 x_{7} & +  0.84 x_{12} & -24.38 x_{4} & -5.83 x_{5} & -0.54 x_{14} & -0.99 x_{13} & +  1.15 x_{11}\\
 x_{2}   &  2.03630911164 & +  0.03 x_{9} & -1.16 x_{7} & -0.06 x_{12} & +  0.80 x_{4} & +  0.45 x_{5} & -0.03 x_{14} & +  0.06 x_{13} & -0.02 x_{11}\\
 x_{3}   &  0.408969506386 & -0.06 x_{9} & +  0.35 x_{7} & -0.03 x_{12} & +  0.13 x_{4} & -0.79 x_{5} & +  0.06 x_{14} & +  0.01 x_{13} & +  0.00 x_{11}\\
 x_{8}   &  1.16996142092 & +  0.05 x_{9} & -1.13 x_{7} & -0.01 x_{12} & +  0.48 x_{4} & -0.27 x_{5} & -0.02 x_{14} & +  0.04 x_{13} & +  0.05 x_{11}\\
 x_{6}   &  0.445307252972 & +  0.03 x_{9} & +  0.13 x_{7} & -0.01 x_{12} & +  1.31 x_{4} & +  0.07 x_{5} & +  0.03 x_{14} & +  0.06 x_{13} & -0.07 x_{11}\\
\hline
z    &  16.8633332119 & +  0.05 x_{9} & -8.87 x_{7} & -0.57 x_{12} & +  8.80 x_{4} & -3.61 x_{5} & +  0.29 x_{14} & +  0.56 x_{13} & -0.40 x_{11}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  2.99900412298 & +  0.04 x_{9} & -0.51 x_{7} & -0.03 x_{12} & -0.03 x_{10} & -0.30 x_{5} & +  0.02 x_{14} & -0.03 x_{13} & +  0.04 x_{11}\\
 x_{4}   &  1.04110766728 & +  0.01 x_{9} & +  0.04 x_{7} & +  0.03 x_{12} & -0.04 x_{10} & -0.24 x_{5} & -0.02 x_{14} & -0.04 x_{13} & +  0.05 x_{11}\\
 x_{2}   &  2.87013080217 & +  0.04 x_{9} & -1.13 x_{7} & -0.03 x_{12} & -0.03 x_{10} & +  0.26 x_{5} & -0.05 x_{14} & +  0.03 x_{13} & +  0.02 x_{11}\\
 x_{3}   &  0.549324693927 & -0.06 x_{9} & +  0.35 x_{7} & -0.02 x_{12} & -0.01 x_{10} & -0.83 x_{5} & +  0.06 x_{14} & +  0.01 x_{13} & +  0.01 x_{11}\\
 x_{8}   &  1.66802340893 & +  0.06 x_{9} & -1.11 x_{7} & +  0.01 x_{12} & -0.02 x_{10} & -0.39 x_{5} & -0.03 x_{14} & +  0.02 x_{13} & +  0.08 x_{11}\\
 x_{6}   &  1.80850443729 & +  0.04 x_{9} & +  0.19 x_{7} & +  0.03 x_{12} & -0.05 x_{10} & -0.25 x_{5} & +  0.00 x_{14} & +  0.00 x_{13} & -0.01 x_{11}\\
\hline
z    &  26.0297400382 & +  0.11 x_{9} & -8.51 x_{7} & -0.26 x_{12} & -0.36 x_{10} & -5.72 x_{5} & +  0.09 x_{14} & +  0.20 x_{13} & +  0.02 x_{11}\\
\end{array}\]


 $ x_{9} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  3.3501598668 & -0.64 x_{3} & -0.29 x_{7} & -0.04 x_{12} & -0.03 x_{10} & -0.83 x_{5} & +  0.05 x_{14} & -0.03 x_{13} & +  0.05 x_{11}\\
 x_{4}   &  1.10132019037 & -0.11 x_{3} & +  0.08 x_{7} & +  0.03 x_{12} & -0.04 x_{10} & -0.33 x_{5} & -0.02 x_{14} & -0.04 x_{13} & +  0.05 x_{11}\\
 x_{2}   &  3.23308137736 & -0.66 x_{3} & -0.90 x_{7} & -0.05 x_{12} & -0.04 x_{10} & -0.28 x_{5} & -0.01 x_{14} & +  0.04 x_{13} & +  0.02 x_{11}\\
 x_{9}   &  9.47350778706 & -17.25 x_{3} & +  6.04 x_{7} & -0.39 x_{12} & -0.10 x_{10} & -14.24 x_{5} & +  0.99 x_{14} & +  0.15 x_{13} & +  0.17 x_{11}\\
 x_{8}   &  2.19335779222 & -0.96 x_{3} & -0.77 x_{7} & -0.01 x_{12} & -0.02 x_{10} & -1.18 x_{5} & +  0.03 x_{14} & +  0.03 x_{13} & +  0.09 x_{11}\\
 x_{6}   &  2.14544932149 & -0.61 x_{3} & +  0.40 x_{7} & +  0.02 x_{12} & -0.06 x_{10} & -0.75 x_{5} & +  0.04 x_{14} & +  0.01 x_{13} & +  0.00 x_{11}\\
\hline
z    &  27.0515257334 & -1.86 x_{3} & -7.86 x_{7} & -0.31 x_{12} & -0.37 x_{10} & -7.26 x_{5} & +  0.20 x_{14} & +  0.21 x_{13} & +  0.04 x_{11}\\
\end{array}\]


 $ x_{11} $ enters and Unbounded Dictionary!


 $ x_{11} $ enters and Unbounded Dictionary!
\end{document}
